# Algorithms in Real Algebraic Geometry by Denis S. Arnon, Bruno Buchberger

By Denis S. Arnon, Bruno Buchberger

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**Example text**

Xm ]. Let us remark that the cluster algebras A(x, B) and A(x, − B) are naturally isomorphic for all extended clusters x and all exchange matrices B, because they have the same exchange relations. An ice quiver is a quiver Q = ( Q0 , Q1 , s, t) without loops and 2-cycles together with a partition of the set of vertices Q0 = M F into two sets, called mutable and frozen vertices, such that the starting and terminating point of an arrow α ∈ Q1 cannot both be frozen vertices. An isomorphism between ice quivers Q = ( Q0 , Q1 , s, t) and Q = ( Q0 , Q1 , s , t ) is an isomorphism ( f 0 , f 1 ) of quivers such that f 0 : Q0 → Q0 maps mutable vertices to mutable vertices and frozen vertices to frozen vertices.

We denote the equivalence class of x = ( x0 , x1 , . . , xn ) ∈ kn+1 \{0} in Pn (k ) by ( x0 : x1 : . . : xn ). In general, note that a d-dimensional vector space is spanned by d linearly independent vectors ( xi1 , xi2 , . . , xin ) ∈ kn for i ∈ {1, 2, . . , d}. The vectors constitute the row vectors of a d × n matrix X of full rank with entries in k. Two such matrices X, Y ∈ Matd×n (k ) define the same subspace if and only if there exists an invertible matrix d × d matrix A such that X = AY.

Now assume that m ≥ 2. Using the induction hypothesis we deduce that, depending on whether m is even or odd, c2n,p,q = c2n−1,p,q + c2n−2,p,q−1 = (n−1q− p)(n−p q) + (n−q−1−1 p)(n−p q) = (n−q p)(n−p q), −q c2n+1,p,q = c2n,p,q + c2n−1,p−1,q = (n−q p)(n−p q) + (n−q p)(np− = (n−q p)(n+p1−q), 1) which proves the statement. The author is not aware of a direct combinatorial proof. Nonetheless the proposition gives an explicit formula for the coefficients that appear when we write the cluster variables as Laurent polynomials in the initial cluster variables.